#### MATH 1610

##### Final - Practice 1 | General
1. Find the derivative of $y = x\sqrt{1 - x^2}$.

A) $\frac{-x}{\sqrt{1 - x^2}}$
B) $\frac{x}{\sqrt{1 - x^2}}$
C) $\sqrt{1 - x^2} - \frac{x^2}{\sqrt{1 - x^2}}$
D) $\sqrt{1 - x^2} + \frac{x^2}{\sqrt{1 - x^2}}$
E) $\sqrt{1 - x^2} + \frac{x^2}{2\sqrt{1 - x^2}}$

2. Evaluate $\lim_{x \to -4} \frac{x^2 + x - 12}{x^2 + 7x + 12}$.

A) DNE
B) $5$
C) $6$
D) $7$
E) $8$
F) $9$
G) $\frac{0}{0}$

3. Evaluate $\int \sin x - \csc^2 x \; dx$

A) $-\cos x - \sec x + C$
B) $-\cos x + \cot x + C$
C) $\cos x - \cot x + C$
D) $-\cos x + \csc x + C$
E) $\cos x - \csc x + C$

4. Evaluate $\frac{d}{dx} \int_{2x}^{3x} \frac{1}{\ln t} dt$

A) $\frac{3}{\ln 3x} - \frac{2}{\ln 2x}$
B) $\frac{3x}{\ln 3x} - \frac{2x}{\ln 2x}$
C) $\frac{3x}{\ln t} - \frac{2x}{\ln t}$
D) $\frac{1}{3x} - \frac{1}{2x}$
E) $\frac{1}{\ln 3x} - \frac{1}{\ln 2x}$

5. Find the deivative of $x^2y^2 + x^3 + y = 8$ at $(2, 0)$.

A) $0$
B) $-3$
C) $-6$
D) $-9$
E) $-12$
F) $-15$

6. $f(x) = e^{2x}. Evaluate$\lim_{h \to 0} \frac{f(0 + h) - f(0)}{h}$. A)$\frac{0}{0}$B) DNE C)$0$D)$1$E)$2$F)$3$7. Find the$x$value of the point on the curve$y = 3x + 2$closest to the point$(1, 0)$. A)$-\frac{3}{2}$B)$-1$C)$-\frac{1}{2}$D)$0$E)$\frac{1}{2}$F)$1$G)$\frac{3}{2}$8. An icicle in the shape of a cone is growing in volume at the rate of$1 \; \text{cm}^3/\text{min}$. The height always equals twice the radius of the base. When the height equals$10$cm, how fast is the height increasing? (Hint:$V = \frac{1}{3} \pi r^2 h$.) A)$\frac{1}{25\pi}$B)$\frac{1}{75\pi}$C)$\frac{1}{225\pi}$D)$\frac{1}{315\pi}$E)$\frac{1}{400\pi}$9. Let$y = xe^x$. Where is$y$increasing? A)$x > 0$B)$x > -1$C)$x > 2$D)$x < -1$E)$x < 0$10. Find the derivative of$y = \cot^{-2}(x)$. A)$2 \cot^{-3}(x) \csc^2(x)$B)$-2 \cot^{-3}(x) \csc^2(x)$C)$2\cot^{-2}(x) \csc(x)$D)$-2\cot^{-2}(x) \csc(x)$E)$\-csc^{-4}(x)$11. Solve$f'(x) = 3x^2 + e^x$for$f(x)$when$f(0) = 2$. A)$f(x) = x^3 + e^x + 1$B)$f(x) = x^3 + e^x + 2$C)$f(x) = 6x + e^x$D)$f(x) = x^3 + e^x$E)$f(x) = 3x^2 + e^x + 2$12. Evaluate$\int_1^2 3x^2 - 2x \; dx$A)$4$B)$3$C)$2$D)$1$E)$0$F)$-1$13. Find the general antiderivative of$3x^{1/2} + x^{-1/2}$. A)$\3x^{3/2} - x^{1/2} + C$B)$2x^{3/2} + 2x^{1/2} + C$C)$2x^{3/2} - 2x^{1/2} + C$D)$\frac{3}{2} x^{-1/2} + \frac{1}{2}x^{-3/2} + C$E)$\frac{9}{2} x^{3/2} - \frac{1}{2}x^{1/2} + C$14. Find the horizontal asymptote for$y = \frac{1 + 7x^2 + 3x^3}{x^4 - x}$. A)$y = -1$B)$y = 0$C)$y = 1$D)$y = 2$E)$y = 3$15. Find the vertical asymptote(s) for$y = \frac{1 + 7x^2 + 3x^3}{x^4 - x}$. A)$x = 0$B)$x = 0, x = 1$C)$x = -1, x = 0, x = 1$D)$x = -1, x = 1$E)$x = 1$16. Find the absolute maximum of the function$f(x) = x^4 - 8x^2$on the interval$-1 \le x \le 3$. A)$-20$B)$-16$C)$-7$D)$0$E)$3$F)$9$G)$12$17. Find the absolute minimum of the function$f(x) = x^4 - 8x^2$on the interval$-1 \le x \le 3$. A)$-20$B)$-16$C)$-7$D)$0$E)$3$F)$9$G)$12$18.$f(x) = x^4 - 6x^2$and$f'(x) = 4x^3 - 12x$. Where is$f(x)$concave up? A)$x > 1$B)$x < -1, x > 1$C)$-1 < x < 1$D)$-\sqrt{3} < x < 0, x > \sqrt{3}$E)$x < -\sqrt{3}, 0 < x < \sqrt{3}$19. Evaluate$\lim_{x \to -4} \frac{x}{(4 + x)^6}$. A)$\frac{1}{0}$B)$-\infty$C)$\infty$D) DNE E) None of the above 20. Evaluate$\int \frac{2x^3}{\sqrt{x^4 + 9}} dx$A)$x^4\sqrt{x^4 + 9} + C$B)$\frac{1}{2} \sqrt{x^4 + 9} + C$C)$\frac{1}{\sqrt{x^4 + 9}} + C$D)$\frac{6x^2\sqrt{x^4 + 9} - 4x^6(x^4 + 9)^{-1/2}}{x^4 + 9} + C$E)$\sqrt{x^4 + 9} + C$21.$y = \frac{x^3 + 3x}{x}$. Find$y''$. A)$0$B)$1$C)$2$D)$3$E)$4$F)$5$G)$6$22. Evaluate$\lim_{x \to 1} \frac{\ln x - x + 1}{(x - 1)^2}$A)$-1/6$B)$-1/4$C)$-1/2$D)$1/2$E)$1/4$F)$1/6$23. Find the derivative of$y = \arctan(x^2)$. A)$\frac{2x}{1 + x^2}$B)$\frac{2x}{1 + x^4}$C)$-\tan^{-2}(x^2)\sec^2(x^2)2x$D)$\frac{1}{1 + x^4}$E)$\tan^{-2}(x^2)\sec^2(x^2)2x$24. Evaluate$\int_0^{1/2} \frac{1}{\sqrt{1 - x^2}} dx$A)$0$B)$\frac{\pi}{6}$C)$\frac{\pi}{4}$D)$\frac{\pi}{3}$E)$\frac{\pi}{2}$25. Find the derivative of$y = \frac{\ln^2 x}{x}$. A)$\frac{2 \ln x - \ln^2 x}{x^2}$B)$\frac{2x \ln x - \ln^2 x}{x^2}$C)$\frac{2 \ln x}{x}$D)$\frac{2 - 2 \ln x}{x^2}$E)$\frac{\frac{1}{x} - \ln^2 x}{x^2}$26. Evaluate$\int \frac{9x^4 + 5x^2}{x^{1/2}} dx$A)$\frac{\frac{9x^5}{5} + \frac{5x^3}{3}}{\frac{2x^{3/2}}{3}} + C$B)$2x^{9/2} + 2x^{5/2} + C$C)$9x^{7/2} + 5x^{3/2} + C$D)$\frac{x^{1/2} (36x^3 + 10x) - (9x^4 + 5x^2) \frac{1}{2}x^{1/2}}{x}$E)$\frac{81}{2}x^{9/2} + \frac{25}{2} x^{5/2} + C$27. Evaluate$\int \frac{3x^2}{\sqrt{1 - x^6}} dx$A)$2 \sqrt{1 - x^6} + C$B)$3x^2 \arcsin (x^3) + C$C)$\arcsin(x^3) + C$D)$\frac{6x\sqrt{1 - x^6} + 9x^7(1 - x^6)^{-1/2}}{1 - x^6}$E)$\ln | \sqrt{1 - x^6} | + C$28. Evaluate$\lim_{x \to 1} \frac{\frac{1}{3 - x} - \frac{1}{x + 1}}{x - 1}$. A) DNE B)$-\frac{1}{4}$C)$\frac{1}{4}$D)$-\frac{1}{3}$E)$\frac{1}{3}$F)$-\frac{1}{2}$G)$\frac{1}{2}$29. Evaluate$\lim_{x \to 3} \frac{3 - \sqrt{12 - x}}{3 - x}$A)$\frac{0}{0}$B) DNE C)$\frac{1}{6}$D)$0$E)$-\frac{1}{6}$F)$\frac{1}{2}$G)$-\frac{1}{2}$30. Which of the following graphs most closely has the following properties: • Increasing$-1 < x < 0, 0 < x < 1, x > 1$. Decreasing$x < -1$. • Concave down$x < -1, -1 < x < 0, x > 1$. Concave up$0 < x < 1$. • Vertical asymptotes$x = -1, x = 1$. Horizontal Asymptote$y = 1$. •$f(0) = 0\$

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